The figures in the margin indicate full marks for the questions.

1. Answer the following questions as directed: (1 × 8 = 8)

• (a) Fill in the blank: ________ is neither a composite number nor a prime number.
• (b) Write True or False: Between any two odd numbers, there always lies an even number.
• (c) Fill in the blank: Statistics is ________ of facts.
• (d) Write True or False: Questionnaire should be accompanied by a covering letter.
• (e) Fill in the blank: Trade discount is charged on ________ price of an article.
• (f) Fill in the blank: Slant surface area of a cone is ________.
• (g) Fill in the blank: There are ________ types of classification.
• (h) Define relative frequency of a class interval.

2. Solve the following questions: (2 × 5 = 10)

(a) Solve: \( |2x - 4| = 7 \)

(b) Evaluate: \( \log_{2\sqrt{7}} 784 \)

(c) Simplify: \( (125y^3 \div 8a^{-3})^{-2/3} \)

(d) Find the fourth proportional to the following numbers: Rs. 8, Rs. 13, 32 kg.

(e) Convert the following number in the form:\( \frac{p}{q} \)

3. Answer the following questions:

(a) Define compound ratio with an example and define 'rule of three'.

(b) If \( \frac{9^n \cdot 3^2 (3^{-n})^{-1} - (27)^n}{3^{3m} \cdot 2^3} \)= \( \frac{1}{27} \), prove that \( m = 1 + n \).

If \( pqr = 1 \), then prove that \( \frac{1}{1+p+q^{-1}} + \frac{1}{1+q+r^{-1}} + \frac{1}{1+r+p^{-1}} = 1 \).

(c) If 1 inch = 25.4 mm, then express 3 miles in terms of km (correct to 2 decimal places).

(d) The parallel sides of a trapezium are 24 cm and 52 cm and the remaining two sides are 30 cm and 26 cm, find its area.

The diagonal of a square floor is 84 m. Find the area of the floor. The floor is to be covered with coir carpet of 1.4 m wide. The cost of carpet is Rs. 16 per meter and the cost of labour is 20% of the cost of carpet. Find the total cost of covering the floor.

(e) Solve any one:

1) \( x^{2/3} - 3x^{1/3} + 2 = 0 \)
2) \( 3(x - \frac{1}{x})^2 - 5(x - \frac{1}{x}) - 8 = 0 \)

Descriptive Questions (5 Marks Each)

4. Answer either (a) or (b): (5)
(a) A cylindrical jar of radius 10 cm contains water to a depth of 12 cm. If 24 spheres each of radius 2.5 cm are immersed in it, the jar will be completely filled with water. Find the height of the jar.
(b) How many meters of cloth 2.2 m wide will be required to make a conical tent, whose vertical height is 12 m and radius of the base is 16 m? Find also the cost of cloth at the rate of Rs. 7 per meter.

5. A trader mixes wheat costing Rs. 16 per kg with wheat costing Rs. 14 per kg and sells the mixture at Rs. 16 per kg, thereby making a profit of 7.5% on selling price. In what ratio does he mix these? (5)

What are the precautions to be taken while using secondary data?

6. What are the precautions to be used while drafting a questionnaire? (5)

7. If the square of the distances between the points (12, 5) and (x, 3) is 29 units, then find the value of x. (5)

8. How many spherical balls of radius 1 cm each can be made out of a solid metallic cube of edge 22 cm? (5)

9. Are (a, 0), (-a, 0) and \( (0, a\sqrt{3}) \) the vertices of an equilateral triangle? Examine. (5)

10. If the points (p + 1, 1), (2p + 1, 3) and (2p + 2, 2p) are collinear, then prove that either \( p = 2 \) or \( p = \frac{1}{2} \). (5)

"Statistics are like clay of which you can make a God or a devil as you please." Discuss.

Section D: Long Answer Questions

11. (a) Prove that: \( (xy)^{\log x - \log y} \times (yz)^{\log y - \log z} \times (zx)^{\log z - \log x} = 1 \) (5)

Prove: \( \frac{1}{1+\log_a bc} + \frac{1}{1+\log_b ca} + \frac{1}{1+\log_c ab} = 1 \)
(b) What is the difference between Primary data and Secondary data? (4)
(c) Define the following (any two): Parameter, Statistic, Sample (2)

12. (a) Write a brief note on different types of diagram used in Statistics. (4)
(b) Represent the following data by a pie diagram: (4)

Head of Expenditure	Expenditure (Rs. lakhs)
Agriculture	8,000
Irrigation	5,000
Industry	3,000
Transport	2,400
Misc.	1,600
Total	20,000

13. (a) Fill in the blanks: (1 × 3 = 3)
i. 1 acre = ________ sq. yards.
ii. The volume of a sphere is ________.
iii. Bar diagram is ________ dimensional diagram.
(b) Write a brief note on different parts of a statistical table. (2)

What are the objectives of classification? Name various types of classification.

14. Following are the weights (kg) of 30 students: (5)

Prepare a frequency distribution table showing tally marks taking the class interval as 5-9, 10-14...... etc. also find:
(a) Frequency density of each class;
(b) Percentage of students whose weight is less than 30 kg;
(c) Percentage of students whose weight is more than 35 kg.